Physics, Special Functions, and Asymptotics

It’s often said that special functions come up in physics. This is certainly true, but how do they come up in physics? This post gives a high-level answer, and the report linked in the post gives more details.

I’ve written several posts lately on Runge-Kutta methods, specifically the system of equations one must solve to design an RK method. The number of equations for an n-step method is equal to the number of distinct rooted trees with up to n nodes.

The asymptotic form for the number of rooted trees with exactly n nodes is known. I derive here the asymptotic form for the number of rooted trees with up to n nodes.